Toeplitz Operators on the Upper Half-Plane with Homogeneous Symbols

Abstract

AbstractIn this chapter we return to the upper half-plane Π, the space L2(Π) and its Bergman subspace A2(Π). Passing to polar coordinates we have $$ L_2 \left( \prod \right) = L_2 \left( {\mathbb{R}_ + ,rdr} \right) \otimes L_2 \left( {\left[ {0,\pi } \right],d\theta } \right) = L_2 \left( {\mathbb{R}_ + ,rdr} \right) \otimes L_2 \left( {0,\pi } \right), $$ and $$ \frac{\partial } {{\partial \bar z}} = \frac{{\cos \theta + i\sin \theta }} {2}\left( {\frac{\partial } {{\partial r}} + i\frac{1} {r}\frac{\partial } {{\partial \theta }}} \right) = \frac{{\cos \theta + i\sin \theta }} {{2r}}\left( {r\frac{\partial } {{\partial r}} + i\frac{\partial } {{\partial \theta }}} \right). $$KeywordsBoundary PointCompact OperatorToeplitz OperatorLocal AlgebrBergman ProjectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.